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You will prove that the Fibonacci numbers satisfy the identity \(F_n^2 + F_{n+1}^2 = F_{2n+1}\text{.}\) One way to do this is to prove the more general identity,

\begin{equation*} F_mF_n + F_{m+1}F_{n+1} = F_{m+n+1}\text{,} \end{equation*}

and realize that when \(m = n\) we get our desired result.

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